Nowadays, with the development and popularization of digital cameras, almost all im- ages obtained are color ones that have a better visual experience and have the ability to convey more information than monochrome or binary images, and thus are applied in numerous fields, from the casual documentation of events to medical applications. However, most of the existing color image processing methods are still the traditional methods of processing monochrome images. Although these traditional methods have achieved great success in processing monochrome images, when processing color im- ages, they either process each color channel independently or handle the concatenation of three color channels using the concatenation model, which all ignore the mutual connection among channels of color images, and thus usually result in color distortion during the processing. As a successful color image representation tool, quaternion has achieved excel- lent results in color image processing, because it treats the color image as a whole rather than as a separate color space component, thus it can make full use of the high correlation among color channels. Compared with traditional methods, the quaternion- based models process three color channels information in a parallel way and thus can preserve the correlation among the color channels well. This thesis mainly studies quaternion-based optimization problems and their applications in color image process- ing. In this regard, we do the following works: The first work is about quaternion-based bilinear factor matrix norm minimiza- tion for color image inpainting. In this work, we propose three novel low-rank quaternion matrix completion (LRQMC) methods based on three quaternion- based bilinear factor (QBF) matrix norm minimization models including quater- nion double Frobenius norm (Q-DFN)-based, quaternion double nuclear norm (Q-DNN)-based, and quaternion Frobenius/nuclear norm (Q-FNN)-based mod- els. The second work is about low-rank quaternion tensor completion for recovering color videos. In this work, we reconstruct a color video as a third-order quater- nion tensor. Under the definition of Tucker rank, the global low-rank prior to quaternion tensor is encoded as the nuclear norm of unfolding quaternion matri- ces. Then, we provide the completion algorithm for any order (≥ 2) quaternion tensors by applying the alternating direction method of multipliers (ADMM) framework. The method theoretically can be well used to recover missing en- tries of any multidimensional data with color structures. The third work is about quaternion higher-order singular value decomposi- tion (QHOSVD) and its applications in color image processing. In this work, motivated by the advantages of the higher-order singular value decomposition (HOSVD) and the quaternion tool, we generalize the HOSVD to the quaternion domain and define QHOSVD. Due to the non-commutability of quaternion mul- tiplication, QHOSVD is not a trivial extension of the HOSVD. They have similar but different calculation procedures. Theoretically, QHOSVD is a proper tensor generalization of the quaternion singular value decomposition (QSVD) and a proper quaternion generalization of the standard HOSVD. From the application point of view, the defined QHOSVD can be widely used in various visual data processing with color pixels. As examples, in this thesis, we present two appli- cations of the defined QHOSVD in color image processing— multi-focus color image fusion and color image denoising.